@UltraMex
Here are some additional changes you may need to make to get the ESO launcher working correctly with Steam.
1. The .exe that Steam requires to be launched to be compatible with the Steam overlay is not actually eso.exe, but a file called "zosSteamStarter.exe". If your Steam library is "pointing" to the wrong .exe file, it won't launch from the library. The correct file "zosSteamStarter.exe" will usually be in the following directory:
Steam\Steamapps\Common\Zenimax Online
After locating "zosSteamStarter.exe", right click it to run it as an administrator. You may want to set it to permanently run as an administrator in your Compatibility tab. You can also create a shortcut to this .exe to use it more easily in the future; this is the .exe that must be run to access the Steam overlay and log your Steam play hours.
2. In order for "zosSteamStarter.exe" to log the information such as play hours properly, you may have to lower your Windows UAC (User Account Control) settings. Lowering User Account Control settings to the minimum will allow Steam to gather this data properly.
You also may be encountering an issue where the ESO Launcher is not being added to the Zenimax Online directory. The thread linked below has troubleshooting which can help resolve this issue.
SOLUTION: Steam Windows 10 "Elder Scrolls Online is preparing to install" crash at 100% downloaded